1. How to Make Standards Operational in theCurriculum Dr. Don S. Balka Saint Mary’s College Notre Dame, Indiana Director, National Council of Teachers of Mathematics Macmillan/McGraw-Hill

2. © 2008 Don Balka. All rights reserved. Education in Saudi Arabia Primary School (6 - 12): Mathematics Intermediate School (12 - 15): Mathematics General Secondary School (15 - 18): Mathematics Technical Secondary School (15 - 18): Vocational/Technical: Mathematics Commercial: Mathematics Agriculture: Mathematics

3. © 2008 Don Balka. All rights reserved. Mathematics in a Changing World Mathematics for life Mathematics as a part of cultural heritage Mathematics for the workplace Mathematics for the scientific and technical community

4. © 2008 Don Balka. All rights reserved. Historical Background All students should learn important mathematical concepts with understanding Standards are critical for those who are decision-makers Curriculum and Evaluation Standards for School Mathematics Professional Standards for Teaching Mathematics Assessment Standards for School Mathematics

5. © 2008 Don Balka. All rights reserved. Principles and Standards for School Mathematics 2000

6. © 2008 Don Balka. All rights reserved. Purpose Set forth a comprehensive and coherent set of goals for mathematics for all students, PK - 12 Serve as a resource for teachers, education leaders, and policymakers to use in examining and improving the quality of mathematics instruction programs

7. © 2008 Don Balka. All rights reserved. Purpose Guide the development of curriculum frameworks, assessments, and instructional materials Stimulate ideas and ongoing conversations at the national, provincial or state, and local levels about how best to help students gain a deep understanding of important mathematics Purpose

8. © 2008 Don Balka. All rights reserved. Principles for School Mathematics Equity Curriculum High expectations and strong support Coherent, focused, articulated TeachingUnderstanding, challenging, supporting Learning Understanding, building new knowledge from experience Assessment Supporting the learning and furnishing information Technology Essential; enhancing student learning

9. © 2008 Don Balka. All rights reserved. Principles and Standards for School Mathematics NCTM Curriculum Principle “…a well-articulated curriculum gives teachers guidance regarding important ideas or major themes, which receive special attention at different points in time. It also gives guidance about the depth of study warranted at particular times and when closure is expected for particular skills or concepts.” (PSSM)

10. © 2008 Don Balka. All rights reserved. Standards for School Mathematics Content Standards Number and Operations Algebra Geometry Measurement Data Analysis and Probability

11. © 2008 Don Balka. All rights reserved. Standards for School Mathematics Process Standards Problem Solving Reasoning and Proof Communication Connections Representation

12. © 2008 Don Balka. All rights reserved. A Framework for Mathematical Proficiency Conceptual understanding: comprehension of mathematical concepts, operations, and relations Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently, and appropriately Strategic competence: ability to formulate, represent, and solve mathematical problems Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification Productive disposition: habitual inclination to see mathematics as sensible, useful, and worthwhile National Research Council, 2001

13. © 2008 Don Balka. All rights reserved. Number of Fourth-Grade GLE per State by Strand

14. © 2008 Don Balka. All rights reserved.

15. What Are Curriculum Focal Points? Concept of a Focal Point Each topic begins in the background of a particular grade as a context for exploration, then emerges into the foreground at a later grade level for intense instruction, and then recedes into the background again.

16. © 2008 Don Balka. All rights reserved. The Content of Curriculum Focal Points Three per grade level, PreK-8 Often represent multiple content strands Describe the majority of instruction for a specific grade level Taken together across grade levels, provide the major components of a mathematically sound, coherent, and cohesive preK- 8 curriculum

17. © 2008 Don Balka. All rights reserved. How Will Curriculum Focal Points Be Used? To design instruction around the question, “What are the most important ideas at my grade level?” To provide information about how ideas at one grade level fit with the important ideas in previous and following grades To prioritize uses of activities, assessments, and other published materials

18. © 2008 Don Balka. All rights reserved.

19. Curriculum Focal Points Algebra and Its Connections GRADE 3 CONNECTIONS Algebra Understanding properties of multiplication and the relationship between multiplication and division is a part of algebra readiness that develops at grade 3 The creation and analysis of patterns and relationships involving multiplication and division should occur at this grade level Students build a foundation for later understanding of functional relationships by describing relationships in context with such statements as, “The number of legs is 4 times the number of chairs.”

20. © 2008 Don Balka. All rights reserved. Hawaii Objectives for Algebra Grade Cluster Benchmarks 2 - 3 Create and describe spatial and numerical patterns State a general rule that describes a given pattern Represent the same pattern in different forms Use concrete, pictorial, and verbal representations of numerical situations Symbolize mathematical situations Quantify comparisons and describe change in familiar situations

21. © 2008 Don Balka. All rights reserved. Arizona Objectives for Algebra PERFORMANCE OBJECTIVES Concept 1: Patterns (PO 1) Communicate a grade-level appropriate iterative pattern, using symbols or numbers (3 - 5) Concept 2: Functions and Relationships (PO 1) Describe the rule used in a simple grade-level appropriate function (K - 8)

22. © 2008 Don Balka. All rights reserved. Arizona Objectives for Algebra PERFORMANCE OBJECTIVES Concept 3: Algebraic Representations (PO 1) Evaluate expressions involving the four basic operations by substituting given whole numbers (decimals, fractions, rational values) for the variable(s) (4 - 6, 8) Concept 3: Analysis of Change (PO 1) Identify the change in a variable over time (1 - 4)

23. EXAMPLES Making Connections in Algebra

24. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections ARROW DIAGRAMS 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

25. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections ARROW DIAGRAMS Go one place to the right! Go one place to the left! Go one place up! Go one place down!

26. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections 34 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

27. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections 68 12 GUESS MY RULE!

28. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections 68 GUESS MY RULE! (0,1), (1,2), (3,4), (99,100) y = x + 1 ADD ONE

29. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections

30. © 2008 Don Balka. All rights reserved. Curriculum Focal Points Algebra and Its Connections HOW MANY WAYS CAN YOU MAKE CHANGE FOR 100 HALALAS USING 5 AND 10 HALALA COINS? An Equation: 5X + 10Y = 100 5 Halala 10 Halala Total 0 10 100 2 9 100 4 8 100 ? ? 100

31. © 2008 Don Balka. All rights reserved. MAKE ME AN L! How many squares are needed for Stage 5? How many squares are needed for Stage 10? How many squares are needed for Stage N? Where is the Algebra?

32. © 2008 Don Balka. All rights reserved. MAKE ME AN L! What do you notice about the number of squares for each stage? Functions! Graphing! f(N) = 2N - 1

33. © 2008 Don Balka. All rights reserved. MAKE ME AN L! What is the total number of squares needed to construct the first 10 stages? 1 4 9 16 Extending the Problem Where’s the Algebra?

34. © 2008 Don Balka. All rights reserved. MAKE ME AN L! Extending the Problem 1 + 3 + 5 + 7 4 x 8 = 32 32 ÷ 2 = 16 = 4 2 For 10 Stages, 10 2 For N Stages, N 2 The sum of the first N odd numbers is N 2. 7 + 5 + 3 + 1 8 + 8 + 8 + 8 How many squares in the first four stages?

35. © 2008 Don Balka. All rights reserved. Skills for the 21st Century Computational Skills Needed by Middle Grades Students Preparing to Live in the 21st Century Intelligent use of calculators, with a clear understanding of their shortcomings Good operational knowledge of basic facts Clear, practical understanding of place value and the system of numeration Understanding of simple fractions, decimals, and percents, including their interrelatedness

36. © 2008 Don Balka. All rights reserved. Skills for the 21st Century Computational Skills Needed by Middle Grades Students Preparing to Live in the 21st Century Self-confidence in, and ability to use, a range of techniques for computing Inclination and the ability to use thinking strategies first when checking calculations, estimating results, or performing various calculations

37. © 2008 Don Balka. All rights reserved. Skills for the 21st Century Computational Skills Needed by Middle Grades Students Preparing to Live in the 21st Century Instinct and willingness to think about numbers in natural, comfortable, and flexible ways when computational results (where exact answers of estimates) are produced. Instinct and willingness to reflect on numerical results so as to judge their reasonableness McIntosh, Reys, and Reys, 1997

38. © 2008 Don Balka. All rights reserved. Considerations Who will write the standards for mathematics? What are the major content strands for the mathematics curriculum? What are the major topics for the mathematics curriculum at each level: Primary, Intermediate, Secondary? Ministry of Education Teachers University Mathematicians Business/Industry Leaders

39. © 2008 Don Balka. All rights reserved. How will professional development for teachers be offered? Nationally, local districts Will the mathematics curriculum include process standards: Problem Solving, Reasoning and Proof, Communication, Connections, Representation? Considerations How will mathematics standards impact teacher preparation for the primary, intermediate, and secondary levels?

40. © 2008 Don Balka. All rights reserved. SUMMARY All students should learn important mathematical concepts with understanding The purpose of standards is to provide a comprehensive and coherent set of goals for mathematics for all students Standards serve as a resource for teachers, education leaders, and policymakers to use in examining and improving the quality of mathematics instruction programs Standards guide the development of curriculum frameworks, assessments, and instructional materials

41. © 2008 Don Balka. All rights reserved. MATHEMATICS CLASSROOM OF THE FUTURE Teachers are empowered. The curriculum is implemented as designed. Multiple instructional strategies are employed. Students are actively engaged. Assessment is varied. Balka, Hull, and Harbin-Miles, 2008

42. Thank you.